{
  "schema_version": 1,
  "problem_number": "KOU-16.60",
  "title": "Elements Inverted by an Automorphism and the Sum of the Character Degrees (Kourovka Notebook Problem 16.60)",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For a finite group G let T(G) be the sum of the degrees of the irreducible complex characters of G, and for an automorphism α of G let S_α = {g ∈ G : α(g) = g⁻¹}. D. MacHale asked in the Kourovka Notebook (Problem 16.60) whether |S_α| ≤ T(G) for every α ∈ Aut G. The inequality is stated as Theorem 3.1 of a 2026 preprint of Yerrapati, Dixit and Shukla; the proof given there for the bound on twisted indicators that it uses (Lemma 2.19 there) is incomplete. We give a complete proof, in two short ways. The first follows the strategy of that preprint: |S_α| = Σ_χ ι_α(χ)χ(1) with ι_α(χ) = |G|⁻¹ Σ_{g∈G} χ(gα(g)), and we prove |ι_α(χ)| ≤ 1 for every automorphism from the orthogonality relations for matrix coefficients. The second restricts α to the subgroup of fixed points of α², on which it induces an automorphism of order at most 2, uses the twisted Frobenius–Schur count of Kawanaka and Matsuyama for involutory automorphisms, and then the inequality T(H) ≤ T(G) for subgroups H. We also show that equality holds if and only if α² = 1 and all twisted indicators are equal to 1; in particular the inequality is strict when α has order at least 3. Special cases were known: involutory automorphisms (from the twisted Frobenius–Schur count), inner automorphisms and complete groups (Bors, 2019), and abelian groups, dihedral groups and groups of order p, 2p, p² (Yerrapati, Dixit and Shukla, 2026). The novelty is modest: a complete proof of the general case and the characterisation of equality. This is an unrefereed note.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.GR",
    "math.RT"
  ],
  "keywords": [
    "finite groups",
    "automorphisms inverting elements",
    "sum of character degrees",
    "twisted Frobenius-Schur indicator",
    "orthogonality relations",
    "Frobenius reciprocity",
    "Kourovka Notebook",
    "Problem 16.60",
    "MacHale",
    "complete answer",
    "UnsolvedMath",
    "KOU-16.60",
    "math.GR",
    "math.RT",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-18",
  "publication_date": "2026-10-18",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-18",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/kou-16-60/",
  "pdf_url": "https://eulersolve.org/papers/kou-16-60/paper.pdf?v=0703c1a976f1",
  "doi": "10.5281/zenodo.23445966",
  "zenodo_record_url": "https://zenodo.org/records/23445966",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The inequality is also stated as Theorem 3.1 of the 2026 preprint arXiv:2605.23195; special cases were known (involutory automorphisms; inner automorphisms and complete groups, Bors 2019). The note gives a complete proof and the case of equality; its novelty is modest. Unrefereed note.",
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}